Find a relationship between 1:n

[c0mrade]

hi guys
any help in this would be deeply appreciated
i am new here
this is not homework. its an assignment?

this is my maths ia and i need help with it
i am asked to find a conjecture

see some of you might have heard about this question.

in an equilateral triangle ABC, a line segment is drawn from each vertex to a point on the opposite side so that the segment divides the side in the ratio 1:2, creating another equilateral triangle DEF
find a relationship between 1:n (like 1:2) and the area of the two triangles

i have done all the working out. i just need a conjecture and how to prove it

 

[c0mrade]

anyone?

 

[LukeD]

Does your name happen to be Kyle?

----

Try just drawing the smaller equilateral triangle inside the larger one. The answer should jump right out at you. If that doesn't help, then think: How long are each of the sides of the new triangle compared to the old one? What do you think this would do to the area? Why?

 

[c0mrade]

no actually
i have done heaps of all that stuff


ABC : DEF
1) Ratio of sides of Triangle3 ABC to DEF(mm) 150 : 98  (1:4)
150/98 = 1.53
2) Ratio of sides of Triangle2 ABC to DEF(mm) 160 : 88  (1:3)
160/88 = 1.82
3) Ratio of sides of Triangle1 ABC to DEF(mm) 180 : 68  (1:2)
180/68 = 2.65
Pattern: For every 1 units of increment in the side of triangle ABC, there is a decrement in the side of triangle DEF.

Ratio of area of Triangle3 ABC to DEF(mm2) 2.34 : 1  (1:4)
Ratio of area of Triangle2 ABC to DEF(mm2) 3.30 : 1  (1:3)
Ratio of area of Triangle1 ABC to DEF(mm2) 7 : 1  (1:2)
Pattern: When tried different things, an obvious pattern that emerges is that, the square root of the fraction of the area of the triangles (ABC divided by DEF) is equal to the fraction of the of ratio of the triangles (ABC divided by DEF).


just need a conjecture

 

[HallsofIvy]

What do you mean by "the ratio of the triangles" if not the ratio of the areas of the triangles? The ratio of corresponding lengths of the triangles? If that is the case you are correct but that does not solve your original problem since you haven't shown that cutting the sides of the original triangle is the ration 1:2 will produce a new triangle with the ratio of its sides to that of the original triangle is 1:2.